Rock Engineering Rock EngineeringPractice & Design Practice & DesignLecture 8:Lecture 8: Stress Analysis aroundStress Analysis aroundStress Analysis aroundStress Analysis around Underground Openings Underground Openings1 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionAuthors Note: Authors Note:The lecture slides provided here are taken from the course Geotechnical Engineering Practice, which is part of the 4th year Geological Engineering program at the University of British Columbia (VC d )Th ki i d(Vancouver, Canada). The course covers rock engineering and geotechnical design methodologies, building on those already taken by the students covering Introductory Rock Mechanics and Advanced Rock Mechanics Rock Mechanics. Although the slides have been modified in part to add context, they of course are missing the detailed narrative that accompanies any l l dh hl lecture. It is also recognized that these lectures summarize, reproduce and build on the work of others for which gratitude is extended. Where possible, efforts have been made to acknowledge thv ri us s urc s ith listf r f r nc s b inpr vid dt th the various sources, with a list of references being provided at the end of each lecture. Errors, omissions, comments, etc., can be forwarded to the 2 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionErrors, omissions, comments, etc., can be forwarded to the author at: erik@eos.ubc.caStress Stress--Controlled Instability Mechanisms Controlled Instability Mechanismsll ll d i bili i lld ib Structurally-controlled instabilities are generally driven by a unidirectional body force, i.e. gravity. Stress-controlled instabilities, however, are not activated by a single force, but by a tensor with six independent componentsHencethea tensor with six independent components. Hence, the manifestations of stress-controlled instability are more variable and complex than those of structurally-controlled failures.3 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionStress Stress--Controlled Instability Mechanisms Controlled Instability MechanismsKaiser et al (2000)4 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionKaiser et al. (2000)Stress Stress--Controlled Instability Mechanisms Controlled Instability MechanismsAl h hhf d ll i fh fhb Although the fundamental complexity of the nature of stress has to be fully considered in the design of an underground excavation, the problem can be initially simplified through the assumptions of continuous, homogeneous, isotropic, linear elastic behaviour (CHILE). homogeneous, isotropic, linear elastic behaviour (CHILE).CHILE: Continuous, Homogeneous, Isotropic, Linear ElasticDIANE: Discontinuous, Inhomogeneous, Anisotropic, Non-ElasticTh i i tii h th l tib d thCHILEThe engineering question is whether a solution based on the CHILE assumption are of any assistance in design. In fact though, many CHILE-based solutions have been used successfully, especially in those excavations at depth where high stresses have closed the fractures andp gthe rock mass is relatively homogeneous and isotropic. However, in near-surface excavations, where the rock stresses are lower, the fractures more frequent, and the rock mass more disturbed and weathered, there is more concern about the validity of the CHILE model 5 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editionmore concern about the validity of the CHILE model.Stress Stress--Controlled Instability Mechanisms Controlled Instability MechanismsA stress analysis begins with a knowledge of the magnitudes and directionsof the in situ stresses in the region of the excavation. This allows for the calculation of the excavation disturbed or induced stresses.wn (2006)There exists several close form solutions for the induced stresses Brady & BrowThere exists several close form solutions for the induced stresses around circular and elliptical openings (and complex variable techniques extend these to many smooth, symmetrical geometries), and with numerical analysis techniques the values of the induced stresses can be determined accurately for any three-dimensional B6 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editionstresses can be determined accurately for any three-dimensional excavation geometry. Stresses & DisplacementsStresses & Displacements - - Circular Excavations Circular ExcavationsThe Kirsch equations are a set of closed-form solutions, derived from the theory of elasticity, used to calculate the stresses and displacements around a circular excavation.pPk P kPStress ratio:Brady & Brown (2006)7 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editionk = h/vStresses & DisplacementsStresses & Displacements - - Circular Excavations Circular ExcavationsFrom these equations we can see that the stresses on the boundary (i.e. when r = a) are given by:= p[(1+k) + 2(1-k)cos2] = p[(1+k) + 2(1 k)cos2]rr= 0 r= 0 h h d l rNote that the radial stresses are zero because there is no internal pressure, and the shear stresses must be zero at t ti n fb und a traction-free boundary.8 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionExample 1: Stresses around a Circular Opening Example 1: Stresses around a Circular OpeningQ. At a depth of 750 m, a 10-m diameter circular tunnel is driven in rock having a unit weight of 26 kN/m3and uniaxial compressive and tensile strengths of 80 0 MPa and 3 0 MPa strengths of 80.0 MPa and 3.0 MPa, respectively.Will the strength of the rock on the tunnel boundary be exceeded if:(a) k=0 3andA(a) k=0.3, and(b) k=2.0?Since the tunnel has neither a support pressure nor an internal Harrison & Hudson (2000)A. Since the tunnel has neither a support pressure nor an internal pressure applied to it, the local stresses at the boundary have 3= r= 0 and 1= . The Kirsch solution for the circumferential stress is:For a location on the tunnel boundary 9 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionFor a location on the tunnel boundary (i.e. a = r), this simplifies to:Example 1: Stresses around a Circular Opening Example 1: Stresses around a Circular Opening1Hudson (2000)First, we can assume that the vertical stress is equal to the weight of the overburden, giving:MP 5 19 750 026 0Harrison & HMPa zv5 . 19 750 026 . 0 2The extreme values of induced stress occur at positions aligned with the principal in situ stresses, and so in order to compute the stress induced in the crown and invert (i.e. roof and floor) we use = 90, and forn wn n n ( . . f n f ) w u 9 , n fthe sidewalls we use = 0.For k=0.3:Crown/invert ( = 90):= -1.95 MPa (i.e. tensile) Sidewalls ( = 0): = 52 7 MPaSidewalls ( = 0 ): = 52.7 MPa For k=2.0:Crown/invert ( = 90): = 97.5 MPaSidewalls ( = 0): = 19 5 MPa compressive strength is 10 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionSidewalls ( = 0 ): = 19.5 MPa strength is exceededStress and Failure Criterion Stress and Failure CriterionThe key is that the failure criterion is compared to the stresses and is not part of the calculation!! Well see this is different when similar relationships based on Mohr-Coulomb and Hoek-Brown are incorporated into stress-strain constitutive relationships. 11 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionBrady & Brown (2006)Orientation ofOrientation of 11& Induced Stresses & Induced StressesPotential Ground Control Issues:Destressing = wedge failuresConcentration = spalling1111d t i t destressingstress t tidestressingstress concentrationconcentrationdestressingStresses can be visualized as flowing around the excavation periphery in the direction of the major principle stress (1)Where they divergerelaxation 12 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editiondirection of the major principle stress (1). Where they diverge, relaxation occurs; where they converge, stress increases occur. Stresses Away from OpeningStresses Away from Opening 13 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionBrady & Brown (2006)Zone of Influence Zone of InfluenceThe concept of influence is important in excavation design, since the presence of a neighbouring opening may provide a significant disturbance to the near-field stresses to the point of causing f l failure. (a) axisymmetric stress distribution around a circular iih d t ti t opening in a hydrostatic stress field; (b) circular openings in a hydrostatic stress field, effectively isolated by virtue offf ct yat yrtu f their exclusion from each others zone of influence.Brady & Brown (2006)14 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionBrady & Brown (2006)Zone of Influence Zone of Influence(2006)dy & Brown (Brad illustration of the effect of contiguous openings of different dimensionsThe zone of influence of excavation I includes excavationdimensions. The zone of influence of excavation I includes excavation II, but the converse does not apply.15 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionStresses Around Elliptical Openings Stresses Around Elliptical OpeningsThe stresses around elliptical openings can be treated in an analogous way to that just presented for circular openingsThere is muchpresented for circular openings. There is much greater utility associated with the solution for elliptical openings than circular openings, because these can provide a firstpapproximation to a wide range of engineering geometries, especially openings with high width/height ratios (e.g. mine stopes, power house cavernsetc ) house caverns, etc.).From a design point of view, the effects of changing either the orientation within the stress field or the aspect ratio of suchstress field or the aspect ratio of such elliptical openings can be studied to optimize stability.16 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionStresses Around Elliptical Openings Stresses Around Elliptical OpeningsAssuming isotropic rock conditions, an elliptical opening is completely characterized by two parameters: aspect ratio (major to minor axis) which is the eccentricity of the ellipse; and orientation withaxis) which is the eccentricity of the ellipse; and orientation with respect to the principle stresses. The position on the boundary, with reference to the x-axis, is given by the angle .Hudson & Harrison (1997)17 of 37 Erik Eberhardt UBC Geological Engineering ISRM Edition( )Stresses Around Elliptical Openings Stresses Around Elliptical OpeningsIt is instructive to consider the maximum and minimum values of the stress concentrations around the ellipse for the geometry of an ellipse aligned with the principal stresses. It can be easilyp g p p yestablished that the extremes of stress concentration occur at the ends of the major and minor axes.Hudson & Harrison (1997)18 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionExample 2: Stresses around an Elliptical Opening Example 2: Stresses around an Elliptical OpeningQ. A gold-bearing quartz vein, 2 m thick and dipping 90, is to be exploited by a small cut-and-fill stoping operation. The mining is to take place at a depth of 800 m, and the average unit weight of the r nite h st r ckb ve this level is 29 kN/m3The strikef thegranite host rock above this level is 29 kN/m3. The strike of the vein is parallel to the intermediate stress, and the major principal stress is horizontal with a magnitude of 37.0 MPa. The uniaxial compressive strength of the vein material is 218 MPa, and the tensile Ap gstrength of the host rock is -5 MPa. What is the maximum permissible stope height before failure occurs.W hi2 Di hi d d iA. We can assume that, in 2-D cross-section, the stresses induced in the sidewalls (tensile) and the crown (compressive) of the stope can be approximated using the equations for an elliptical excavation.19 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionHarrison & Hudson (2000)Example 2: Stresses around an Elliptical Opening Example 2: Stresses around an Elliptical Opening Rearranging the given equations, we can solve for the height of the excavation as the minimum of:1Hudson (2000)or Harrison & HThe maximum stress that can be sustained by the crown and the sidewall are 218 and 5 MPa, respectively. Note that the sidewall stress is negative because this represents the tensile strength2stress is negative because this represents the tensile strength.3The vertical stress is:and hence the ratio of horizontal to vertical stress is:20 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionExample 2: Stresses around an Elliptical Opening Example 2: Stresses around an Elliptical OpeningThe maximal height of a stope such that the compressive strength of the rock in the crown is not exceeded is given by:4The maximal height of a stope such that the tensile strength of the rock in the sidewall is not exceeded is given by: (2000)ison & Hudson Thus we see that sidewall failure is the limiting condition if no stress-induced failure is 21 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionHarrf no str ss n ucfa ur s acceptable in the stope design.Zone of Influence Zone of Influence elliptical approximation to the zone of influence around an elliptical ti excavation.22 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionHudson & Harrison (1997)Stress AnalysisStress Analysis -- Numerical Modelling Numerical ModellingMany underground excavations are i l i h irregular in shape and are frequently grouped close to other excavations other excavations. These problems require the use of numerical techniques numerical techniques. 23 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionEberhardt et al. (1997)Numerical Modelling Numerical ModellingNumerical methods of stress and deformation analysis fall into two categories:Integral Methodsincl. boundary-element methodonly problem boundary is defined & discretizedPro: more computationally efficient Con: restricted to elastic analyses Differential incl. finite-element/-difference & distinct-element methodsproblem domain is defined &D fferent al Methodsproblem domain is defined & discretizedPro: non-linear & heterogeneous material properties accommodated Cl l titi24 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionCon: longer solution run times Boundary Boundary--Element & Stress Analyses Element & Stress AnalyseshB dEl h dThe Boundary Element Method (BEM) is generally favoured for stress analyses involving multiple excavations withmultiple excavations with complex 3-D geometries (e.g. those frequently encountered in underground mine design). The i l h f th irregular shape of the orebodies and staged nature of mining, makes the ease of mesh generation andgcomputational efficiency afforded by the BEM highly advantageous. Eberhardt et al (1997)Complex 3-D geometries: Its easier to generate a mesh over a surface than through a volumeEberhardt et al. (1997)Examine3D (Rocscience) - http://www.rocscience.com/Map3D (Mine Modelling Pty Ltd.) - http://www.map3d.com/ BEFE (Computer Software & Services - CSS) - http://members.chello.at/sylvia.beer/ Commercial Software:25 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionGPBEST (Best Corp.) - http://www.gpbest.com/BEASY (Beasy Group) - http://www.beasy.com/Boundary Boundary--Element & Stress Analyses Element & Stress AnalysesI f i l i hb d fh In performing an analysis, the boundary of the excavation is divided into elements and the interior of the rock mass is represented mathematically as an infinite continuum. Themathematically as an infinite continuum. The solution works to find a set of approximate stresses which satisfy prescribed boundary conditions, and then uses these to calculate the t d di l tith k stresses and displacements in the rock mass.What to Know: Computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in boundary integral form). Key advantage is the reduction of the model dimension by one, providing simpler mesh generation & input data preparation, and greater computational efficiency. Key disadvantage is the required assumption of homogeneous linear elastic material behaviour; plasticity and heterogeneity negate the methods intrinsic simplicity. Common applications include stress analysis of underground excavations, soil-structure interactionsbrittle fracturing processesdynamic problems26 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editionstructure interactions, brittle fracturing processes, dynamic problems, groundwater flow and coupled H-M & T-H-M problems.Boundary Boundary--Element & Stress Analysis Element & Stress AnalysisExamine2D)cscience E27 of 37 Erik Eberhardt UBC Geological Engineering ISRM Edition(RocCase History: Trout Lake Mine,Case History: Trout Lake Mine, Flin Flin Flon FlonSince 1927, Hudson Bay Mining & Smelting has developed more than 29 mines in the Flin Flon Greenstone belt, with continuous production of zinccoppersilverand goldproduction of zinc, copper, silver, and gold since 1930. 28 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionCase History: Trout Lake Mine,Case History: Trout Lake Mine, Flin Flin Flon FlonThe Trout Lake Cu-Zn sulphide deposit was discovered in 1976, with mining beginning in 1982. The deposit is a VMS-type deposit that involves two ore zones 500 m apart, 29 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editionthat involves two ore zones 500 m apart, which reach depths exceeding 1200 m.Flin Flin Flon Flon Mining Belt Mining BeltProjected life of mine. of mine.In a historic mining centre like Flin Flon, the life of the town is dependent on the ability to mine new and deeper reserves to feed the mill and keep the smelter running mill and keep the smelter running.30 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionCase History: Trout Lake Mine,Case History: Trout Lake Mine, Flin Flin Flon FlonInitial mining was carried out by ramping down to theInitial mining was carried out by ramping down to the deposit and mining it at relatively shallow depths through sub-level stoping. To what depths, though, will mining be economical? rown (2006)Brady & Br31 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionCase History: Trout Lake Mine,Case History: Trout Lake Mine, Flin Flin Flon Flon (1997)rhardt et al. EberGround controldilution and drift stability 32 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionGround control, dilution and drift stability problems encountered at 400m depth.Case History: Trout Lake Mine,Case History: Trout Lake Mine, Flin Flin Flon Flon (1997)rhardt et al. 33 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionEberCase History: Trout Lake Mine,Case History: Trout Lake Mine, Flin Flin Flon FlonIf rock mass failure is a problem at shallower depths, what will happen when the mine d d d ? expands and goes deeper?34 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionCase History: Trout Lake Mine Stress Analysis Case History: Trout Lake Mine Stress AnalysisR schk& R m n ski (1993)Empirical analysis based on performance of stope backs in the Flin Flon deposits, although useful, may be limited in its applicability if hd l dh Reschke & Romanowski (1993)the database is populated with cases at shallower depths (i.e. lower stresses).Eberhardt et al (1997) Eberhardt et al. (1997)2-D and 3-D stress analyses of 35 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionyTrout Lake stopes.Case History: Trout Lake Mine Stress Analysis Case History: Trout Lake Mine Stress AnalysisBecause of the efficiency of the boundary element technique, 3-D analyses (e.g. those looking at the stress interactions between neighbouring stopes)interactions between neighbouring stopes) becomes a relatively straightforward exercise.Eb h d l (199 )pillarEberhardt et al. (1997)Computed stresses can then be compared against estimated rock mass strengths to 36 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editionagainst estimated rock mass strengths to assess pillar stability.Lecture References Lecture ReferencesBrady, BHG & Brown, ET (2006). Rock Mechanics for Underground Mining (3rdEdition). Chapman &Hall: London.Eberhardt, E, Stead, D, Reeves, MJ & Connors, C (1997). Design of tabular excavations infoliated rock: An integrated numerical modelling approach. Geotechnical and Geological Engineering 15(1): 47-85.Harrison, JP & Hudson, JA (2000). Engineering Rock Mechanics Part 2: Illustrative WorkedExamples. Elsevier Science: Oxford.Hudson JA & Harrison JP (1997) Engineering Rock Mechanics An Introduction to the Principles Hudson, JA & Harrison, JP (1997). Engineering Rock Mechanics An Introduction to the Principles .Elsevier Science: Oxford.Kaiser, PK, Diederichs, MS, Martin, D, Sharpe, J & Steiner, W (2000). Underground works inhard rock tunnelling and mining. In GeoEng2000, Melbourne. Technomic Publishing Company:Lancaster pp 841 926 Lancaster, pp. 841-926.37 of 37 Erik Eberhardt UBC Geological Engineering ISRM Edition